Optimal. Leaf size=49 \[ \frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2 x-\sqrt{3}+3\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+3\right ) \]
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Rubi [A] time = 0.0155641, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {632, 31} \[ \frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2 x-\sqrt{3}+3\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+3\right ) \]
Antiderivative was successfully verified.
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Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x}{3+6 x+2 x^2} \, dx &=\frac{1}{2} \left (1-\sqrt{3}\right ) \int \frac{1}{3-\sqrt{3}+2 x} \, dx+\frac{1}{2} \left (1+\sqrt{3}\right ) \int \frac{1}{3+\sqrt{3}+2 x} \, dx\\ &=\frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (3-\sqrt{3}+2 x\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (3+\sqrt{3}+2 x\right )\\ \end{align*}
Mathematica [A] time = 0.02062, size = 44, normalized size = 0.9 \[ \frac{1}{4} \left (\left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+3\right )-\left (\sqrt{3}-1\right ) \log \left (-2 x+\sqrt{3}-3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 31, normalized size = 0.6 \begin{align*}{\frac{\ln \left ( 2\,{x}^{2}+6\,x+3 \right ) }{4}}+{\frac{\sqrt{3}}{2}{\it Artanh} \left ({\frac{ \left ( 6+4\,x \right ) \sqrt{3}}{6}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40655, size = 55, normalized size = 1.12 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3} + 3}{2 \, x + \sqrt{3} + 3}\right ) + \frac{1}{4} \, \log \left (2 \, x^{2} + 6 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6719, size = 136, normalized size = 2.78 \begin{align*} \frac{1}{4} \, \sqrt{3} \log \left (\frac{2 \, x^{2} + \sqrt{3}{\left (2 \, x + 3\right )} + 6 \, x + 6}{2 \, x^{2} + 6 \, x + 3}\right ) + \frac{1}{4} \, \log \left (2 \, x^{2} + 6 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09926, size = 46, normalized size = 0.94 \begin{align*} \left (\frac{1}{4} - \frac{\sqrt{3}}{4}\right ) \log{\left (x - \frac{\sqrt{3}}{2} + \frac{3}{2} \right )} + \left (\frac{1}{4} + \frac{\sqrt{3}}{4}\right ) \log{\left (x + \frac{\sqrt{3}}{2} + \frac{3}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05922, size = 62, normalized size = 1.27 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (\frac{{\left | 4 \, x - 2 \, \sqrt{3} + 6 \right |}}{{\left | 4 \, x + 2 \, \sqrt{3} + 6 \right |}}\right ) + \frac{1}{4} \, \log \left ({\left | 2 \, x^{2} + 6 \, x + 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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